Then x = sin θ                                                  …(2)

From (1) putting the value of θ in (2), we get,

Illustrations

We know that y = sin x means y is the value of sine of angle x if we consider domain and co-domain both as set R of real numbers. Sine ratio as seen from the fig. is many-one into function.

ü        But it is clear that if we restrict the domain to [-Π/2 , Π/2]  and range to [–1, 1], then. y = sin x is one-one onto and hence it is invertible.

So, y = sin x                             x ε [-Π/2 , Π/2] , y ε [–1, 1]

Þ  x = sin^–1 y                           y ε [–1, 1] , x ε [-Π/2 , Π/2]

ü        This value of x is called the principal value, i.e. belonging to [-Π/2 , Π/2]  and [-Π/2 , Π/2] range and it is called principal value range.

ü        The smallest numerical angle is called principal value.

ü        In general the inverse circular functions with their domain and range can be as given below:

Inverse Circular Function

Þ            sin^-1 x = θ        iff           sin θ = x, -Π/2 ≤ θ ≤ Π/2

Domain [-1,1]

Range [-Π/2 , Π/2]

Graph

Inverse Circular Function

Þ            cos^-1 x = θ   iff          cos θ = x, 0 ≤ θ ≤ Π

Domain [–1, 1]

Range [0, Π]

Graph

Inverse Circular Function

Þ            tan^-1 x = θ               iff   tan θ= x,  -Π/2< θ < Π/2

Domain (–∞, ∞)

Range (-Π/2 , Π/2)

Graph

Θ =Π/4 ,5Π/4 ,  9Π/4 etc. Now, direct trigonometric functions follow the definition of a function. But in inverse trigonometry, if we say that to a certain value of the trigonometric ratio there correspond many values of the angle, it violates the definition of function as it becomes a one – many relation. That is why, some restrictions have been imposed on the angles, and these are based on the principle values of the angle.

PRE-REQUISITE

ü        If A and B are two non–empty sets then a function from A to B associated to each element x in A, a unique element f(x) in B.

f : A → B

Þ            The set A is called Domain of f.

Þ            The set B is called the co-domain of f.

Þ      The range of f is the set consisting of all the images of the element of the domain A.

Þ      If          Range of f = {f(x) : x ε A}  then the function is onto.

Þ      One-One function: If x₁, x₂ ε A then f(x₁) = f(x₂) → x₁ = x₂

Þ            Contra positively x₁ ≠ x₂ ε f(x₁) ≠ f(x₂)

Þ      Many one function: f(x₁) = f(x₂)
where x₁ ≠ x₂.

Þ      Onto function: If f(A) = B i.e.
Range = co-domain then the function is onto.

Þ      Into function: If f(A) ς B the function is into

Þ      A function is invertible iff it is a one-one onto function. The inverse of a function is defined as if y = f(x),       x ε A, y ε B and f(x) is one-one and onto in A then            x = f ^–1(y) y ε B, x ε A

Þ      If A and B are domain and range of f(x) then B and A are those
of f^–1(x).